1
general::
Close to, on the brink of, as in I was on the verge of calling the doctor when he suddenly got better, or Sara was on the verge of tears when she heard the news. This term uses verge in the sense of "the brink or border of something." [Mid-1800s]
American Heritage Idioms
2
general::
Phrase(s): on the verge of doing something [and] on the verge of something
at the very beginning of doing something; just about to do something. • Bill was on the verge of leaving town when he found a job. • Susan was on the verge of laughter, so she left the lecture hall.
McGrawhill's American Idioms And Phrasal Verbs